Biholomorphic Mapping on the Boundary I

We present a new proof of Chern-Ji's mapping theorem on a strongly pseudoconvex domain with differentiable spherical boundary. We show that a proper holomorphic self mapping of a strongly pseudoconvex domain with the real analytic boundary is biholomorphic. We shall show that a bounded domain D is biholomorphic to an open ball B n+1 whenever the boundary bD is locally biholomorphic to the boundary of an open ball B n+1. Theorem 1. Let D be a simply connected bounded domain in C n+1 with differen-tiable spherical boundary bD. Suppose that there is a biholomorphic mapping φ ∈ H (U ∩ D) ∩ C 1 U ∩ D for a connected open neighborhood U of a point p ∈ bD satisfying φ (U ∩ bD) ⊂ bB n+1. Then the mapping φ is analytically continued to a biholomorphic mapping from D onto B n+1. Our result is a new proof of a weaker version of Chern-Ji's mapping theorem [CJ]. The main steps of our proof come as follows: We show that the inverse mapping φ −1 is analytically continued on the unit ball B n+1 to be a locally biholomorphic mapping ϕ : B n+1 → D. We show that the mapping ϕ is a proper holomorphic mapping onto a universal covering Riemann domain over D. Thus the mapping ϕ is a biholomorphic mapping whenever D is simply connected. We shall study on a proper holomorphic mapping φ between strongly pseudo-convex bounded domains D, D ′ with real analytic boundaries bD, bD ′. Theorem 2. Let D, D ′ be strongly pseudoconvex bounded domains in C n+1 with real analytic boundaries bD, bD ′ and φ : D → D ′ be a proper holomorphic mapping. Then the mapping φ is locally biholomorphic. If D = D ′ , then the mapping φ is a biholomorphic self mapping. Our result is a new proof of a weaker version of Pinchuk's mapping theorem [Pi]. The main steps of our proof come as follows: We show that the mapping φ is analytically continued along any path on bD as a locally biholomorphic mapping when bD is nonspherical so that the mapping φ : D → D ′ are locally biholomorphic. From the study of Theorem 1, we show that the same is true when bD is spherical so that the mapping φ : D → D ′ are locally biholomorphic. For the case …